Unbalanced optimal transport (UOT) extends classical optimal transport to measures with different total masses, but statistical guarantees for Monge-type estimation remain limited. We study unbalanced transport with quadratic cost and Kullback-Leibler marginal penalties and argue that the natural population target is not a map alone, but a transport-growth pair. Consequently, we develop two estimators for the transport-growth pairs under several setups: an optimal transport plan-based estimator for a general case, and a kernel-based estimator for a case with smooth densities. We also show that an error of the estimator achieves the minimax optimal rate by deriving a matching lower bound of the minimax risk. Our main technical contribution is a value-based stability reduction that converts perturbations of the UOT objective into transport and growth risks through a UOT gap condition. These results provide a statistical foundation for Monge-type estimation in unbalanced optimal transport.